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Denormalized Numbers
Instead of using only normalized numbers and allowing this small
gap around 0, PowerPC processor-based Macintosh computers
use denormalized numbers, in which the leading implicit bit, b
0
,
of the significand is 0 and the minimum exponent is used.
The use of denormalized numbers makes the following statement
true for all real numbers:
x-y=0 if and only if x=y
Another advantage of denormalized numbers is that error
analysis involving small values is much easier without the gap
around zero (Demmel 1984). The computer determines that a
floating-point number is denormalized (and therefore that its
implicit leading bit is interpreted as 0) when the biased exponent
field is filled with 0Õs and the fraction field is nonzero.
Infinities
An Infinity is a special bit pattern that can arise in one of two ways:
■ When an operation (such as 1/0) should produce a
mathematical infinity, the result is an Infinity.
■ When an operation attempts to produce a number with a
magnitude too great for the numberÕs intended floating-point
data type, the result might be a value with the largest possible
magnitude or it might be an Infinity (depending on the current
rounding direction).
These bit patterns (as well as NaNs, introduced next) are
recognized in subsequent operations and produce predictable
results. The Infinities, one positive and one negative, generally
behave as suggested by the theory of limits. For example:
■ Adding 1 to +° yields +°
■ Dividing -1 by +0 yields -°
■ Dividing 1 by -° yields -0
The computer determines that a floating-point number is an
Infinity if its exponent field is filled with 1Õs and its fraction field
is filled with 0Õs. So, for example, in single format, if the sign bit
is 1, the exponent field is 255 (which is the maximum biased
exponent for the single format), and the fraction field is 0, the
floating-point number represented is -°.
iMalc Manual
Technical Considerations
Technical Considerations
38